PCA Advanced Examples & Applications

Transcription

1 PCA Advanced Examples & Applications Objectives: Showcase advanced PCA analysis: - Addressing the assumptions - Improving the signal / decreasing the noise

2 Principal Components (PCA) Paper II Example: Ainley, D.G. et al. (2005). Objective: Relate densities of the 12 most abundant species of seabirds to 12 habitat variables: 5 biological, 4 oceanographic, 3 geographic (spatial) 82.3%

3 Principal Components (PCA) Paper II Oceanographic variables examined: sea-surface temperature / salinity, thermocline depth / strength Date Distance to Fronts Chl Max Acoustic Biomass

4 Principal Components (PCA) Paper II Data Manipulations To Avoid Biases: Densities log-transformed to meet normality assumptions Nevertheless, residuals generated in the regressions for some species did not meet those assumptions (Skewness / Kurtosis Test for Normality of Residuals, p < 0.05) Least-squares regression analysis (ANOVA), however, is a very robust procedure with respect to non-normality (Seber, 1977, Kleinbaum et al., 1988) Yet, while these analyses yield the best linear unbiased estimator in the absence of normally distributed residuals, p- values near 0.05 must be viewed with caution (Seber, 1977)

5 Principal Components (PCA) Paper II To avoid double-absences: Only 15-min transects in which any given species was recorded were analyzed The total sample size for the 12 species was 1209 Is this an adequate sample size? Rule of thumb: 5 samples per variable (Tabachnick and Fidell 1989) 1209 / 12 ~ 100 samples per variable

6 Principal Components (PCA) Paper II Analysis Methods: Principal components analysis (PCA), in combination with Sidak multiple comparison tests, used to assess differences in habitat selection among 12 seabird species To test for significant differences in habitat affinities among seabird species, used two one-way ANOVAs: In the first, tested for differences among PC1 scores of each species; in the second, compared the PC2 scores Differences between two species significant if either one or both PC scores differed significantly

7 Principal Components (PCA) Paper II Community-Wide Result: First and second PC axes explained 60% of variance in distribution of 12 species

8 Principal Components (PCA) Paper II Species-specific Results: Species mapped onto two (independent) dimensions Pair-wise associations (tested) denoted by circles Near Fronts Zoop Prey Salty, Green Fish Prey

9 Principal Components (PCA) Comparisons Number of Axes: - Selected 2 easy to interpret (Ainley et al. 2005) - Selected 6 based on eigenvalues > 1 (Weichler et al. 2004) Display of Results: - Plot & table of eigenvalues (Ainley et al. 2005) - Eigenvalues & interpretation (description) (Weichler et al. 2004) Significance Tests: - Pairwise species comparisons (ANOVA) (Ainley et al. 2005) - Correlations with selected variables (Weichler et al. 2004)

10 Influence of Distances in PCA PCA seeks the strongest patterns, with the largest distances: Remember: Outliers distort the real patterns in the data, by adding large distances

11 Mind the PCA Assumptions Because it uses linear combinations of response variables to create the axes, PCA is subject to the assumptions of linearity in the relationships among the variables. Implicit assumption of linearity in relationships between the responses and gradients represented by ordination axes. Ordination axes uncorrelated (by definition, orthogonal). Assumptions met by relatively homogeneous environmental datasets with few zeroes, and rarely met by species datasets. Difficulties when PCA used on zero-rich species datasets because they usually violate normality and linearity assumptions(e.g., high skewness, difficult to normalize).

12 Reducing Noise / Improving Signal Therefore, critical to determine if a strong influence by dominant species / variables is consistent with the underlying assumptions and with your analysis objectives. Alternatively, you can think about what steps are needed to reduce their influence and to meet assumptions. Options for reducing the influence of dominant species involve: standardizing data to eliminate abundance differences deleting highly dominant / rare species creating subsets of variables to analyze separately

13 Standardizing the Data Relativization re-scales all of the data at once, using a common criterion / standard. When it is done by columns (e.g., species), variation across plots is retained, but variation across species is standardized. When its done by rows (e.g., plots), variation across species is retained, but variation across plots is standardized. Sums:

14 Data Relativizations in PC-ORD Relativization by Maximum: When relativization by maximum is set for columns, each cell in a column is divided by the maximum value in the column, replacing absolute values with proportions of the maximum observed value across all sample units. This relativization approach is used when the maximum observed value of a given response across all sample units is considered the maximum potential abundance for that response in this population-species. The relativized values for each response represent proportions of their maximum potential. They are applied to species data to equalize the influence of common / rare species and of abundant / nonabundant species.

15 Data Relativizations in PC-ORD Relativization by Maximum: (input: x > 0; output: from 0 to 1) Divides each cell s value by the total (for the given row or the given column) such that values range from 0 to 1. Sums: Sums:

16 Data Relativizations in PC-ORD General Relativization: General relativization by column totals reduces influence of responses with high total abundance relative to those with low total abundance, because observations are proportional to their intra-response total abundance. This retains the variation in abundances across sample units, but reduces the influence of very common species and increases the influence of rare species. NOTE: General relativization by row totals is applicable if your question of interest focuses on giving each sample the same influence. HINT: this is not reasonable when the columns have different variables

17 Data Relativizations PC-ORD Y X NOTE: p can take on two values: 1 or 2 Remember: City Block vs Euclidean

18 Data Relativizations in PC-ORD General Relativization: (input: x > 0; output: from 0 to 1) If p = 1, Relativization is by column (or row) totals. Appropriate for city-block distance measure (e.g., Sorensen). If p = 2, Relativization is by column (or row) totals. Appropriate for Euclidean distance measure (square root).

19 General Relativizations General Relativization: (by totals) makes area under each species distribution response curve = 1 By columns generalized: (p = 1): By columns generalized: (p = 2): Sums:

20 Other Data Relativization PC-ORD Deviations: Value Mean Z scores: (Value Mean) / SD Binary response: Above (1) / Below (0) Ranks: Assigns ranks (e.g., 0, 0, 6, 9 would receive the ranks 1.5, 1.5, 3, 4)

21 Data Relativization Remember Note: Need to accept TEMP file

22 Relativizations Recommendations Do not use relativization by maximum when any data < 0 Do not use general relativization when any data < 0 Cannot use standard deviates with empty data groups (rows / columns) - Why not? NOTE: Fine to use with negative data

23 PCA Example Upwell Where do we start? Data Exploration + Summarization What do we look for? Value Ranges Typos, Possible Transformations Unequal Sums Different Weights

24 What do we look for? PCA Example Upwell -1 < Skewness < +1 Few Vacant Cells

25 PCA Example Upwell How can we solve unequal sums (weights) of variables? Relativize by Maximum (Columns)

26 PCA Example Upwell How can we solve unequal sums (weights) of variables? Standard Deviates by Columns

27 Mind your Relativizations Not all datasets amenable to all relativizations: some are mathematically incompatible, others fail to relativize the samples / species. Check Data Ranges / Sums BEFORE Check Data Ranges / Sums AFTER

28 Rotating the Ordination PCA seeks the strongest patterns, with the largest distances: The resulting ordination can be rotated to look at specific patterns

29 Axis 2 PCA Tools Rotation Results: Rotation (VARIMAX) Samples: Not rotated Varimax rotated points Species: Axis 2 Eup Poapra Broine Agrrep Cardra Eup Broine Agrrep Poapra Cardra Agrsto vectors Desces Agrsto Desc es Axis 1 Axis 1 Comparison of ordination of sample units in species space before and after varimax rotation. Note the improved alignment of the species vectors with the ordination axes in the rotated ordination.

30 PCA Tools - Rotation Rotation to align patterns from separate ordinations facilitates comparisons across studies: In Ordination 1, the point cloud has been rotated to maximize loading of Variable 1 onto Axis 1. In Ordination 2, the same dominant trends were found but at an angle to those found in Ordination 1. Therefore, Ordination 2 can be rotated through an angle (shown by arrow) so that it aligns Variable 1 with Axis 1.

31 PCA Tools Rotation Rotation aligns ordination to highlight certain patterns NEDO Axes Loadings Axis 1: Axis 2: Rotation by NEDO Stretch plot along direction of most variation for species

32 PCA Tools Rotation Looking at a Specific Species Response Correlations NEDO Axis 1: Axis 2:

33 PCA Tools - Rotation Rotation aligns ordination to highlight certain patterns NOTE: loadings of the species on the axes and the correlations of the species with the axes will change after rotation is implemented

34 Mind your Rotations Report all rotations in results. Check Axis Correlations / % Variance BEFORE Check AxisCorrelations / % Variance AFTER

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36 PCA Next Steps Example 1 Use PCA to synthesize cross-correlated environmental variables into independent (orthogonal) patterns Use new synthetic variables to compare categorical variables (groups) using ANOVA / GLMs

37 PCA Example of Next Steps Principal Component Analysis (PCA) used to assess patterns of shared variation in 71 POP analytes. 6 DDTs,47 PCB congeners, 8 chlordane isomers, 3 hexachlorohexanes dieldrin, mirex, aldrin, hexachlorobenzene, and 10 PBDE congeners. Considered three categorical variables: Three age / sex groups compared in the analysis: juveniles, adult males, and adult females. Two sample origins: necropsy (dead) / biopsy (alive). Two tissues sampled: serum (blood) and fat.

38 PCA Example of Next Steps Sample Outliers: Data log transformed and examined to determine the existence of outliers (> 3 S.D. deviations from mean). Two adult male outliers (one high and one low) were removed for statistical analysis following these criteria. Empty Variables: POP analytes that were below LOQ in > 75% of samples removed to reach recommended 5:1 sample / variable ratio

39 PCA Example of Next Steps Significant PCA axes selected using alpha = 0.05, using 999 randomizations. One significant PCA axis accounted for 74.89% of variance.

40 PCA Example of Next Steps Sample loading values compared using ANOVA to assess whether common patterns of POP levels associated with different age/sex groups (juvenile, adult, adult ), origin (live biopsy vs. necropsy), or tissue (blubber vs. serum). No significant differences among age/sex groups in PCA loading values, indicating that shared variation of POPs did not differ between age/sex groups. Significant difference between the 2 sample origins (p = 0.02), suggesting a difference in POPs between necropsy and live animal samples. Significant difference between blubber samples and serum samples (p < 0.001).

41 PCA Next Steps Example 2 Use PCA to synthesize cross-correlated environmental variables into independent (orthogonal) patterns Use new synthetic variables to explain other response variables (like species counts) using other statistical methods (GLMs, GAMs)

42 PCA Next Steps Published Example: Ainley & Hyrenbach (2010). Objective: Relate seabird densities to five crosscorrelated environmental variables: MEI, PDO, upwelling 39, upwelling 36, SST

43 PCA Next Steps Objective: Also considered lagged environmental data: winter, early spring, late spring

44 PCA Next Steps Results: Four PC axes described 83 % of variability Assessed temporal trends in PC factors using Spearman rank correlations (df = 19, rs critical = 0.433). Tests indicated no trends in spring-time environmental conditions sampled during the study period: PC1 (rs = 0.195, 0.50 > p > 0.20), PC2 (rs = , 0.50 > p > 0.20), PC3 (rs = 0.005, p > 0.50), and PC4 (rs = , p > 0.50).

45 PCA Next Steps Results: Related seabird densities to 4 PC factors and time using GLM tests: - R squared - P value - # of variables

46 PCA Next Steps Results: Species with significant responses to PC1

47 Summary Next Steps 1 PCA synthesized complex patterns into orthogonal axes Other statistical tests performed with resulting PC loadings This allows performing categorical comparisons (i.e., ANOVA)

48 Summary Next Steps 2 PCA synthesized complex patterns into orthogonal axes Other statistical tests performed with resulting PC loadings This allows relating species abundances (non-normal data) to the PCA factors using other statistics (i.e., GLMs, GAMs)